ISBN 978-0-620-44584-9
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ALGORITHMIC DETERMINATION OF POLE-ZERO
REPRESENTATIONS OF POWER TRANSFORMERS’ TRANSFER
FUNCTIONS FOR INTERPRETATION OF FRA DATA
M. Heindl1*, S. Tenbohlen1, A. Kraetge2, M. Krüger2, J. L. Velásquez2
Universität Stuttgart, Institute of Power Transmission and High Voltage Technology, Germany
2
OMICRON electronics GmbH, Austria
*Email: maximilian.heindl@ieh.uni-stuttgart.de
1
Abstract: Frequency Response Analysis compares measured transfer functions of power
transformers. Deviations of frequency response curves indicate electrical or mechanical damages
of windings. As assessments are done by experts, no objective guidelines for interpretation of
measurement results exist. This paper deals with approximation of measured power transformer
frequency responses using complex rational function models. A fitting algorithm maps the
information contained in measured curves on a pole-zero model of reduced complexity. The aim is
to develop an algorithm for automated interpretation using analytical models created on the basis
of measurement data. Requirements of accuracy of the approximation models for purposes of
Frequency Response Analysis are considered. An enhanced algorithm for complex rational
function estimation based on the proven Vector Fitting method is presented. Finally, application of
the algorithm with real measured data is demonstrated.
1.
test TF, which e.g. has been measured when the
transformer was offline during service.
No or slight differences between TFs indicate no
electrical or mechanical change inside the transformer
while deviations suggest potentially critical variances
in relation to the reference. Figure 1 shows an
illustration of a power transformer seen as an electrical
multi-port network.
INTRODUCTION
Frequency Response Analysis (FRA) is one of the key
tools in power transformer condition assessment
enabling detection of winding and core faults. Its
application is mostly meaningful after events such as
electrical faults in the connected power grid or after
transport of a power transformer. Changes of the
transfer function (TF) indicate mechanical and/or
electrical changes of the active part. However,
interpretation of frequency response deviations is a
non-standardised step and is therefore not objective.
Mathematic modelling of power transformer transfer
functions obtained by measurements is – in this
approach – regarded to be the first step for future
automatic interpretation of frequency response
deviations since assessments can be based on a set of
rules evaluating the changes of two analytical transfer
function models rather than the measured data.
1.1.
Iout,1
Iin
RLC Network
Uin
Iout,2
.
.. U
Iout,n
Uout,1
out,2
Uout,n
Figure 1: Power transformer as multi-port network.
Particularly three phase transformers offer numerous
possible (complex) transfer functions. Regarding one
quadripole, two main types are transferred voltage
Basic principle of FRA
The electrical behaviour of power transformers in
higher frequency range is determined through a
complex network composed of the resistance and
inductance of the windings as well as various
capacitances between windings, core and tank.
Capacitances play a dominating role in higher
frequencies while core effects can only be seen in
lower frequency range. Geometrical changes of the
windings like displacements, buckling or conductor
tilting are changings of distances between energized
elements which result in altered capacitances.
Moreover, a windings’ self-inductivity can vary due to
altered flux path structure. So it can be said that a
power transformers’ frequency response is a unique
representation of its structural layout.
,
(1)
and input current (input admittance):
,
(2)
Concerning one dedicated two-port network,
investigations in the past revealed, that all TF types
show sensitivity towards mechanical changes [1].
1.2.
State of the art and interpretation difficulties
In the past, FRA experts were mainly dedicated to
work on reliable measurement techniques that ensure
reproducible measurement results. This is an
indispensable premise for FRA ensuring its
FRA is a comparative method: The functional principle
is to compare two complex transfer functions with each
other. One is the so-called reference TF, another is the
Pg. 1
Paper D-26
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
authenticity as a diagnostic method (see [2]). However,
there’s a gap in one’s knowledge concerning the
interpretation of TF deviations.
A visual comparison of the of TF magnitude curves
| TF(f) | done by experts is state of the art. As yet, only
very few rudiments for algorithm assisted
interpretation of FRA results exist [3]. Application of
these approaches with FRA data measured in the field
showed algorithmic shortcomings which confirms that
FRA assessments by experts are not replaceable so far.
1.3.
2.2.
In [4], an iterative method called “Vector Fitting” (VF)
is described which, given the predefined degree of (4),
tries to find the best fitting rational function for a
measured complex frequency response in a least square
sense. The algorithm starts with an arrangement of N/2
stable complex conjugate pole pairs equally distributed
over the whole measured frequency range and relocates
them towards poles of the measured curve within one
iteration step to improve the fitting. The algorithm
reformulates this nonlinear approximation problem by
sequentially
solving
two
linear
problems.
Figure 2 shows a measured TF magnitude curve
together with the magnitudes of the approximation
function after 10th iteration step. The red dashed-dotted
and blue dotted vertical lines show the corresponding
frequency positions of poles after one respectively 10
iteration steps.
FRA interpretation methodologies
Saying interpretation, i.e. (visual) comparison of power
transformer frequency responses, special attention to
resonance characteristic of TFs has to be paid.
Damping differences covering the whole frequency
range are typical for variances in the measurement
setup. Noticeably frequency confined damping
differences, relocation of resonance peaks, creation of
new resonances or disappearance of resonances in
relation to the reference TF are quite evident for
electrical or mechanical changes of the active part of a
power transformer. The challenge of future automatic
FRA interpretation is to capture these assessment
methodologies. Only with algorithmic interpretation,
objective evaluation of FRA measurement results is
possible. Information about resonance behaviour of
TFs lies within a mathematical formulation of
frequency responses. The main aim is to compare
analytical models of frequency responses in order to
grasp deviations of resonances between FRA curves,
which is pre-condition for advanced automatic
interpretation. Further paragraphs of this paper deal
with the analytical formulation of TF by means of an
enhanced rational approximation method.
2.
2.1.
0.7
|TF(f)| / p.u.
0.5
0
1
⎯⎯⎯⎯⎯⎯⎯ 1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Parameters of approximation process
The final result of the approximation process is not
independent from user input parameters. The number
of iterations and the assumed degree of measured TF
play an important role for the algorithm. The degree of
the rational function has to be chosen carefully.
Although the algorithm converges if the chosen ordinal
number N exceeds the needed theoretically needed
complexity, the result will contain lots of unimportant
poles (with great negative real part) which makes it
intransparent. In another word: If a measured
frequency response of e.g. 1000 points is fitted by a
rational function of degree 600, the result is useless.
On the other hand, if the chosen degree is below the
needed value, the algorithm will only deliver poor
accuracy, which is not acceptable. Both parameters –
ordinal number and number of iterations – influence
each other. High numbers of iterations do not improve
the fitting result necessarily, as shown later in this
paper. Additionally, not only ordinal number but also
prior choice of starting pole locations has an effect on
the final fitting result.
(3)
(4)
1.0
Figure 2: Measured transformer frequency response
and approximated functions with corresponding pole
locations of different iteration stages.
1
0.5
Frequency f / MHz
2.3.
Pole location after 1st
iteration step
0.3
0
0
where s denotes the complex frequency of a Laplace
transformed function in time domain. An equivalent
expression to (3) is given by:
0.4
0.1
An analytical function is a mathematical formulation
using a closed expression. TFs of RLC two-port
networks are linear systems and can be described by
rational functions consisting of two polynomials with
real coefficients (am, bn):
. . .
. . .
Pole location after 10th
iteration step
0.2
Analytical modelling of TFs
1
Measured
Fitted
0.6
APPROXIMATION OF FREQUENCY
RESPONSES
0
Rational approximation by Vector Fitting
1
As (am, bn) are real, zeros and poles (zm, pn) come in
complex conjugate pairs.
Pg. 2
Paper D-26
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
3.
measurements exceed the RMSE value of 0.1 but still
can be regarded as acceptable.
FITTING ACCURACY REQUIREMENTS
In this paper, the aim of frequency response
approximation is, to obtain a fitted rational function of
minimal complexity that fulfils the requirements in
accuracy for FRA curve comparisons. The approach of
parameterisation of TF curves for further interpretation
purposes will only be successful, if different algorithms
produce the same pole-zero representations for the
same input data [5]. The need to establish measures
along with threshold values assuring approximation
quality is evident. Loss of information during fitting
process is not acceptable.
3.1.
3.2.
Not only has the parameterization of measured TF to
meet the needs concerning the integrated measure of
RMSE, but also the precision of reproducing resonance
peaks has to be satisfied. When comparing two
frequency response curves, differences of resonance
peaks in magnitude, Q and – above all – frequency are
important. So as to create rational functions for FRA
interpretation to this end, variances of resonances
between measured and fitted curve have to be smaller
than deviations caused by the measurement
respectively resolution of the data recording format. As
an example, Figure 4 shows the result of an
approximation in the low frequency part of a TF. This
fitting result can be regarded as limit case regarding
accuracy.
Measuring of approximation accuracy
Quality of approximation can easily be quantified with
the measure of root mean square error (RMSE) of
standardized TFs:
RMSE
1
·Δ
·Δ
2
(5)
Quality of resonance fitting
0.7
1
0.6
using
Measured
Fitted
1
(6)
·Δ
|TF(f)| / p.u.
0.5
1
0.4
0.3
0.2
Standardization cares for comparability of transfer
function
types
with
different
co-domains.
Investigations with many measured transformer
frequency responses revealed, that approximations with
calculated RMSE in the per mill range deliver
sufficient accuracy. The crucial criterion is, that the
fitting error is smaller than the deviations caused by
measuring tolerances of repeated FRA measurements.
The following figure shows two FRA curves of the
transferred voltage (phase 1W-2W) of the same
transformer.
0.1
0
0
20
30
40
50
60
70
Frequency f / kHz
Figure 4: Measured and approximated TF in lower
frequency range. Small deviations in frequency of
resonances at 65 kHz and 72 kHz can be seen.
A slight shift between resonances at 65 kHz and
72 kHz can be seen. Analysis of many frequency
responses – reference and repeated measurements –
recorded with equipment of different manufacturers
revealed that, depending on measuring parameters like
frequency range, number of recorded frequency points,
linear or logarithmic distribution of points, shifts of
2...3kHz are within the range of resolution respectively
repeatability imprecision. Figure 5 again points out,
that fitting quality has to be evaluated by RMSE as
well as a comparison of the quality of resonance match.
0.7
Reference (1999)
Repeated (2005)
0.6
0.5
|TF(f)| / p.u.
10
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency f / MHz
Figure 3: Reference and repeated measurement of
transferred voltage on phase W of 200MVA
transformer (220kV-110kV) with RMSE = 0.0344.
The time period between reference and repeated
measurement is almost 6 years. Calculated RMSE is
0.0344 which indicates very good repeatability; Other
Pg. 3
Paper D-26
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
0.7
with
Measured
Fitted
0.6
,
2
⁄2
(8)
1 |TF(f)| / p.u.
0.5
0.01 ·
,
0.4
,
(9)
(n ∈ [1, ..., N/2] for f > 0 denotes one half of
N complex conjugate pole pairs).
0.3
0.2
Figure 6 illustrates the convergence problem
encountered with several measured frequency
responses. While accuracy in lower frequency range is
good, it is insufficient for higher frequencies. Even for
high numbers of degree, fitting quality could not be
improved. One would think that by increasing the
number of iteration steps the fitting result can be
improved. Unfortunately this is not an option.
0.1
0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency f / MHz
Figure 5: Measured and fitted transfer function with
RSME = 0.002
Figure 7 shows the development of RMSE of the
frequency response of Figure 6.
Although RMSE is 0.002 in this case, the two
resonances around 920 kHz of the measured curve are
not satisfyingly captured by the approximated curve.
Automatic recognition of this problem instead of visual
check can be challenging. An automatic recognition
algorithm for this problem is in early stage of
development.
0.12
Measured
Konventional Fitting
0.1
|TF(f)| / p.u.
0.08
4.
ENHANCED FITTING ALGORITHM FOR
FRA INTERPRETATION PURPOSES
The Vector Fitting algorithm described in [4] is
meanwhile a proven method for creation of mathematic
models of measured frequency responses. Applications
go from transmission line modeling to extraction of
stray parameter functions for dimensioning of filters
for electromagnetic compatibility purposes. There are
differences between applications concerning frequency
range as well as accuracy. As described in previous
paragraph, requirements for FRA interpretation
considering accuracy are relatively high. This section
describes the adaption of the algorithm to improve
fitting results.
4.1.
0.04
0.02
0
0
·
,
1.0
1.5
2.0
2.5
3.0
Figure 6: Measured TF and approximation using
conventional Vector Fitting. Deviations are obvious.
RMSE stagnates or even increases if pole relocation is
repeated – even with very high degree of the fitting
rational function. This makes clear that an illconditioned starting pole distribution does not lead to
satisfying fitting results.
Convergence – and therefore fitting quality – of the VF
algorithm depends on start conditions. There are two
main points to consider: A good (pre-)estimation of the
degree of the rational fitting function and a suitable
start distribution of poles. Conventional VF requires
the desired number N/2 of complex conjugate pole
pairs as input parameter. The frequency locations of
starting poles are by default distributed evenly over the
entire frequency range fmax = L·Δf of the measured TF:
,
0.5
Frequency f / MHz
Conventional Vector Fitting
,
0.06
0.014
Order N=136
RMSE / p.u.
0.012
0.01
0.008
0.006
0.004
0.002
(7)
0
0
10
20
30
40
50
60
Number of Iterations
Figure 7: RMSE vs. number of iterations.
Pg. 4
Paper D-26
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
This problem can be overcomed by algorithmic
estimation of ordinal number and optimization of
starting pole distribution.
4.2.
Divide TF(f) into W sections
(on the basis of local minima)
Degree estimation of fitting function
For all sections i
The idea behind this approach uses the fact that every
pole creates a local resonance peak in |TF(f)|.
Amplitude and Q of local maximum are not only
determined by αn = Re{pn} but also by adjacent poles
that superpose each other. In fact, not every pole
creates its corresponding local maximum. However,
there exists a certain number of poles between local
minima in |TF(f)|. By segmentation of the frequency
range of the measured TF at frequencies with local
minima, rough estimation of the needed ordinal
number can be achieved. Figure 8 shows the
segmentation of a measured TF.
Fitting Order N := 0
N := N+2
Execute Vector Fitting (Order N)
RMSE
0.05?
Save determined poles of section i
Execute Vector Fitting using determined
poles as starting poles to approximate TF(f)
Figure 9: Structogram of enhanced fitting algorithm.
0.8
TF
Local Minimum
0.7
It displays the compactness of the algorithm which can
be implemented and reproduced easily.
0.6
|TF(f)| / p.u.
W
0.5
5.
COMPARISON OF FREQUENCY
RESPONSES
0.4
Figure 10 shows two measured frequency responses
recorded logarithmically (800 points). There are
obvious deviations beginning at low frequencies.
0.3
0.2
0.1
0
0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Frequency f / MHz
kHz
-10
-20
|TF(f)| / dB
Figure 8: Segmentation of TF using local minima
The precise number of poles between local minima
cannot be identified this way but will be determined
within the next step. For frequency responses
containing mentionable amounts of noise, smoothing
of the TF curve using a simple moving average filter
helped to find appropriate segmentations.
Reference
Test
-30
-40
-50
-60
4.3.
Algorithmic determination of starting poles
-70
0
Because several poles can be located between
frequencies of minima, the exact number and location
of these poles must be determined for each section.
This is done by piecewise executing of the vector
fitting algorithm: For every section of TF(f) with
fmin, k ≤ f < fmin, k+1, a fitting rational function is
determined, beginning with 2nd order. RMSE is
computed after wise. If RMSE < 0.05 is achieved for
this section, the found poles are saved in a table. If
RMSE > 0.05, the degree of the rational function for
this section is increased by 2 and so on, until an
appropriate rational function is found for every section.
As the final step, Vector Fitting is executed using the
determined poles of each section as starting poles. A
Nassi-Schneiderman diagram of the algorithm is shown
in Figure 9.
0.2
0.4
0.6
0.8
1.0
Frequency f / MHz
Figure 10: Two deviating frequency
(800 points, linear-logarithmic view).
responses
A rational function was created for the first frequency
response (reference) with accuracy of RMSE < 0.03.
After that, a second rational fitting function for the test
TF was created using the poles of the first fitting as
starting poles. During this process, no new poles are
created. The Vector Fitting algorithm relocates these
starting poles in order to achieve the most precise
approximation to the second TF in a least square sense.
Figure 11 shows the fitting result of both frequency
responses.
Pg. 5
Paper D-26
Proceedings of the 16th International Symposium on High Voltage Engineering
c 2009 SAIEE, Innes House, Johannesburg
Copyright °
ISBN 978-0-620-44584-9
0.9
6.
Reference Measured
Reference Fitted
Test Measured
Test Fittted
0.8
0.7
There is a need for algorithms helping to assess FRA
measurements. The aim is to find an objective way to
compare measured frequency responses of power
transformers.
The advantage of a mathematical formulation of a TF
over an array of hundreds of measured function values
is the reduction of complexity.
Feasibility of approximation of measured frequency
responses by rational functions is shown. The
presented algorithm is based on the proven Vector
Fitting method.
Mechanics of complex function approximation are
predicated on relocation of poles of a predefined
rational fitting function.
Loss of information for the calculated model of a TF is
not acceptable. Accuracy of the fitting result has to be
in the range of reproducibility imprecision of
measurements. RMSE is a suitable measure for the
quality of a fitting result and should be in the per mill
range in order to fulfil accuracy requirements. Main
resonances in measured TF should be captured with a
frequency deviation of less than the recorded frequency
resolution.
The procedure of TF approximation with Vector
Fitting is improved by pre-estimation of the needed
model complexity along with optimized starting pole
distribution.
The developed algorithm was demonstrated using
measured FRA data. Future interpretation algorithms
may incorporate comparisons of pole patterns of fitted
analytical models.
|TF(f)| / p.u.
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1.0
Frequency f / MHz
kHz
Figure 11:
Reference
and
Test
TF
approximations (800 points, linear-linear view)
with
The advantage of this approach is that slight
differences between TF are easily measurable because
corresponding poles are recognisable in an objective
way: Shifts of resonances can be read directly from a
diagram
like
shown
in
Figure 12. It illustrates distribution of discrete
resonance frequencies over the whole frequency range.
The red curve reflects the reference TF and is therefore
a straight line. The blue curve corresponds with the test
TF. The green curve displays absolute frequency
deviations of resonances.
Poles frequency f / MHz
1.0
0.9
0.8
Reference
Test
7.
0.7
REFERENCES
0.6
[1] J.
Christian,
"Erkennung
mechanischer
Wicklungsschäden mit der Übertragungsfunktion",
Dissertation, Universität Stuttgart, Shaker Verlag,
2002, ISBN 3-8322-0480-6G.
[2] R. Wimmer, S. Tenbohlen, K. Feser, A. Kraetge,
M. Krüger, J. Christian, „The influence of
connection and grounding technique on the
repeatability of FRA-results”, in Proc. of 15th
International Symposium on High Voltage
Engineering, T7-522
[3] R. Wimmer, S. Tenbohlen, M. Heindl, A. Kraetge,
M. Krüger, J. Christian, „Development of
Algorithms to Assess the FRA”, in Proc. of 15th
International Symposium on High Voltage
Engineering, T7-523.
[4] B. Gustavsen and A. Semlyen, "Rational
approximation of frequency domain responses by
vector fitting", IEEE Transactions on Power
Delivery, vol. 14, no. 3, pp. 1052-1061, July 1999.
[5] “Mechanical-Condition
Assessment
of
Transformer Windings Using Frequency Response
Analysis (FRA)”, Cigré Working Group A2.26
Technical
brochure
2008,
ref
#342
0.5
0.4
0.3
0.2
0.1
0
80
Poles Δ f (f) / kHz
CONCLUSION
60
40
20
0
-20
-40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Frequency f / MHz
Figure 12: Comparison of frequency responses using
pole frequencies of fitted rational function models.
Interpretation of slight deviations between frequency
responses is the most challenging task of FRA. The
found analytical representations are a first starting
point for further algorithms contributing to automatic
assessment of FRA measurements.
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